1. Balanced Binary Search Trees
CS-2851 Dr. Mark L. Hornick

2. For any balanced binary search tree, searching, inserting and deleting have worstTime(n) that is logarithmic in n
But insertions and deletions can result in a tree that is highly unbalanced
O(log2 n) performance drifts toward O(n)
A BST is balanced only if if its height is logarithmic in n such that
H = 1+Math.floor(log2(n))
N=8, h= 1+floor(log2(8))
N=5, h > 1+floor(log2(5) )
CS-2851 Dr. Mark L. Hornick

3. An AVL tree is a balanced BST that either is empty or in which…
Heights between left and right subtrees differs by only 1
The left and right subtrees are also AVL trees.
CS-2851 Dr. Mark L. Hornick

4. Another (balanced) AVL Tree20 30 10 35 5 32
CS-2851 Dr. Mark L. Hornick

5. More AVL Tree examples
CS-2851 Dr. Mark L. Hornick

6. Non-AVL Tree examples
CS-2851 Dr. Mark L. Hornick

7. Balance can be maintained through rotations
Rotation: an adjustment to the tree, around an element, that maintains the required ordering of elements
CS-2851 Dr. Mark L. Hornick

8. For any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x.
CS-2851 Dr. Mark L. Hornick

9. CS-2851 Dr. Mark L. Hornick

10. For any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x.
CS-2851 Dr. Mark L. Hornick

11. Exercise: Rotate right about 20 Rule: for any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. 20 35 10 5 12 7
CS-2851 Dr. Mark L. Hornick

12. Solution: Right rotation about 2010 35 10 20 5 5 12 35 12 7 7
Note improvement in height
CS-2851 Dr. Mark L. Hornick

13. Exercise: Rotate right about 10 Rule: for any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. 20 35 10 5 12 7
CS-2851 Dr. Mark L. Hornick

14. Solution: Right rotation about 1020 20 5 35 10 35 10 5 12 12 7 7
Note NO improvement in height
CS-2851 Dr. Mark L. Hornick

15. Exercise: Rotate left about 10 Rule: for any left rotation around element x, the left subtree of x’s right child becomes the right subtree of x. 20 35 10 12 5 11 13 7
CS-2851 Dr. Mark L. Hornick

16. Solution: Left rotation about 1020 20 35 12 10 35 10 13 5 12 5 11 11 13 7 7
Note increase in height
CS-2851 Dr. Mark L. Hornick

17. In a rotation around x, the only restructuring is to the subtree rooted at x (nothing above x is affected)
CS-2851 Dr. Mark L. Hornick

18. Application of rotations in a certain way maintains the balance of Binary Search Trees
Resulting in
AVL trees
Red/Black trees
others…
CS-2851 Dr. Mark L. Hornick

19. Rotations summary
CS-2851 Dr. Mark L. Hornick